Equivalently, the random variable of the F-distribution may also be written

where s12 and s22 are the sums of squares S12 and S22 from two normal processes with variances σ12 and σ22 divided by the corresponding number of χ2 degrees of freedom, d1 and d2 respectively.

In a Frequentist context, a scaled F-distribution therefore gives the probability p(s12/s22 | σ12, σ22), with the F distribution itself, without any scaling, applying where σ12 is being taken equal to σ22. This is the context in which the F-distribution most generally appears in F-tests: where the null hypothesis is that two independent normal variances are equal, and the observed sums of some appropriately selected squares are then examined to see whether their ratio is significantly incompatible with this null hypothesis.

The quantity X has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of σ12 and σ22.[8] In this context, a scaled F-distribution thus gives the posterior probability p(σ22/σ12|s12, s22), where now the observed sums s12 and s22 are what are taken as known.